Simultaneous Pairwise Linear Structural Relationships

Prompted by an example in the medical field we are led to consider linear structural relationships between all pairs of a set of p(>2) random variables Xi(i = 0, ... , p 1). This model has relevance in assessing the relative calibrations and relative accuracies of a set of p instruments, each designed to measure the same characteristic, on a common group of individuals. In such a situation we need to estimate the slope and intercept parameters in the linear structural relationships, and the error variances, as measures of calibration and accuracy, respectively, for the p instruments. In contrast to the case p = 2, no problem occurs of unideiltifiability of the parameters and the maximum likelihood approach is feasible. This is illustrated for the case p = 3, but rather simpler estimators are presented for p > 3, together with their asymptotic standard errors. The case p = 3 may be viewed as a new and useful form of instrumental variable approach to the difficult p = 2 situation. The theoretical results are used in an example involvilng different types of instrumenlts for measuring human lung capacity.